Counting. Combinations. Permutations. September 15, Permutations. Do you really know how to count?

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Clara Dixon
5 years ago
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September 15, 2016 Why do we learn to count first? How is this used in the real world?

Counting In how many unique ways can these five simple objects be arranged?

In real life, how important is this skill? What is counting these ipods really about?

identical elements stationary elements Hang Ups You have been given the job of hanging

Combinations Permutation: the order of the events is important and it matters which

Ranking, matching or sequencing elements or groups to specific identities or positions

Combination: the order of the events doesn t matter and it does not matter which

1 September 15, 2016 Why do we learn to count first? How is this used in the real world? Do you really know how to count? Counting In how many unique ways can these five simple objects be arranged? Combinations Which password is stronger? By how much? hayden hayden! In real life, how important is this skill? What is counting these ipods really about? arranging all tree diagrams factorials arranging some repetition sum or product rule identical elements stationary elements Hang Ups You have been given the job of hanging two pictures on the wall: What about three? Combinations Permutation: the order of the events is important and it matters which item is placed first. Ranking, matching or sequencing elements or groups to specific identities or positions is also a permutation. Combination: the order of the events doesn t matter and it does not matter which item is placed first. -arranging all elements 3 -arranging some elements -repetition of elements -sum or product rule -separate cases Are both ways the same? Permutation: the order of thewhy events arematter important does this again? and it matters which item is placed first. -identical elements -stationary elements -direct & indirect method

How else can we visualize this? Systematic approaches What about 5 pictures? How about 8? What problems are we going to start to run into? Examine the Tree Diagram again.

2 How else can we visualize this? Systematic approaches What about 5 pictures? How about 8? What problems are we going to start to run into? Examine the Tree Diagram again. Factorials A more elegant way Wait. What if you have 5 pictures, but only want to hang 3 of them?! How many ways can you hang 5 pictures? How many ways can you hang 8 pictures?

n P r = n! (n-r)! Hang 5 Hang 8 Hang 3 of 8 pictures 1. Suppose there are eight students that are running for class president (Adam, Bob, Christine, Darlene, Emmett, Francis, Greg and Helen).

3 n P r = n! (n-r)! Hang 5 Hang 8 Hang 3 of 8 pictures 1. Suppose there are eight students that are running for class president (Adam, Bob, Christine, Darlene, Emmett, Francis, Greg and Helen). They each have the opportunity to give a brief speech. How many different orders can they speak in? 2. How many ways can you arrange the letters in the word Factor? 3. How many ways can Joe order four different textbooks on the shelf of his locker? Hang on. What if you have an unlimited supply of replicas for each of these Da Vinci's, and you want to make an arrangement of 3 of them? How many arrangements are possible? 4. A twelve-volume library of different books numbered from 1 to 12 is to be placed on a shelf. How many out-of-order arrangements of these books are there? What in life might this be like? 5. In a particular business, everyone has a three-letter designation after their name. What is the smallest number of people employed by the business if there must be at least two people with the same three-letter designation?

6. What are the chances of your boyfriend guessing the combination of your lock below? Four different books and fourteen different pens are sitting on a table. One of each is selected.

4 6. What are the chances of your boyfriend guessing the combination of your lock below? Four different books and fourteen different pens are sitting on a table. One of each is selected. How many ways are there to make your decision? favourable outcomes total outcomes P(A) = n(a) n(s) successes trials wins games Or = +? And = x First, a little probability 7. Mei is trying to choose a new phone number and needs to choose the last four digits of the number. Her favourite digits are 2, 5, 6, 8, 9. Each digit can be used at most once. a) How many permutations are there that would include four of her favourite digits? b) How many of these would end with the digit 2? c) How many of these would be odd? Can 5 and 9 occupy the last space at the same time?

5 10b). What are the chances of correctly guessing your password if one knows a website requires a 6-8 character password? 8. Ontario's license plates use to be three letters and three numeric digits. How many more license plates were possible after the government added the fourth letter? 9. Which password is stronger? By how much? ridge irhs! 10a). What is your evil step-sister's chance of guessing them and reading your s? What else would help you come up with a more certain answer? How many ways are there to select a prize if your teacher says you can have a pen, a book, or both? How many five-letter "words" can be formed by re-arranging the letters in the word B E E T S

6 In how many ways can all the letters of the word CANADA be arranged if the consonants must always be in the order in which they occur in the word itself? How might this be like the "BEETS" question? 11. Find the number of arrangements of the following words: a) Definitive b) Mississippi c) Mississauga d) Canada 15. In how many ways can a student answer a true-false test that has six questions. Explain your reasoning. 16. The final score of a soccer game is 6 to 3. How many different scores were possible at half-time? 12. Construct a tree diagram to illustrate the possible contents of a sandwich made from white or brown bread, ham, chicken, or beef, and mustard or mayonnaise. How many different sandwiches are possible? 13. In how many ways can you roll either a sum of 4 or a sum of 11 with a pair of dice? 17. A large room has a bank of five windows. Each window is either open or closed. How many different arrangements of open and closed windows are there? 14. A five-digit password must be selected from 26 letters (not-case sensitive), 10 numeric digits and 10 symbols. If you are required to use at least one of each and you may use a digit more than once, how many passwords are possible? Explain your reasoning.

18. A Canadian postal code uses six characters. The first, third, and fifth are letters, while the second, fourth and sixth are digits. A U.S.A. zip code contains five characters, all digits.

7 18. A Canadian postal code uses six characters. The first, third, and fifth are letters, while the second, fourth and sixth are digits. A U.S.A. zip code contains five characters, all digits. a) How many codes are possible for each country? b) How many more possible codes does the one country have than the other? 19. How many 5-digit numbers are there that include the digit 5 and exclude the digit 8? Explain your solution. You already have the skills to answer this, but is there an easier way than counting all of the numbers that qualify? 20. How many... c) How many of the three digit numbers are even numbers and begin with a 4? a) ways can you arrange the letters in the word Boulton? d) How many of the three digit numbers are even numbers and do not begin with a 4? b) Ismail arrange four different textbooks on the shelf in his locker? c) different ways can Laura colour 4 countries on a map if she has a set of 12 coloured pencils and doesn't wish to repeat any? e) Is there a connection among the four answers above? If so, state what it is and why it occurs. d) Five-letter "words" could come before about in the dictionary? 21. Wayne has a briefcase with a three digit combination lock. He can set the combination himself, and his favourite digits are 3, 4, 5, 6 and 7. Each digit can be used at most once. a) How many permutations of three of these five digits are there? 22. After their training run, six members of a track team split a bag of assorted doughnuts. How many ways can the team share the doughnuts if the bag contains a) six different doughnuts? b) three each of two varieties? c) two each of three varieties? b) If you think of each permutation as a three-digit number, how many of these numbers would be odd numbers?

8 23. How many 7-digit even numbers less than 3,000,000 can be formed using all the digits 1, 2, 2, 3, 5, 5, 6? Answer Clues 1. 40, ,001, , ~0.156% 7.a) 120 b) 24 c) ,400, answers vary 10a). answers vary 10b). one in 6,161,227,014,611,136 11a) 302,400 b) 1,247,400 c) 415,800 d) ,912, a). b). 17,476, ,816 20a) 2,520 b) 24 c) 11,880 d) 27,579 21a) 60 b) 36 c) 3 d) 21 e) yes 22a) 720 b) 20 c)

Organized Counting 4.1

Organized Counting 4.1 4.1 Organized Counting The techniques and mathematical logic for counting possible arrangements or outcomes are useful for a wide variety of applications. A computer programmer writing software for a game